you change your league to an "Invitational" League and don't invite them back.

Put in a league rule where a bowlers current average can't fall below their entering average without a majority vote of the captains.

But that would need to be done at the beginning of the league probably since the captain of the offending team would not agree to the rule change.

Pin drop rule is a good solution, but as stated this would have to be voted on by the board.  Or you could adopt a rule where your averages don't re-establish until after a certain amount of weeks, we use 3 weeks in our men's league, you could make it more weeks where it would penalize them for not coming close to their averages.

Either way both suggestions would have to be voted in by the league

I've been through this before.  Its a long story, so I won't go into the whole thing.  In the end the USBC won't do anything.  You can bring them up for dismissal, and have a league vote.  (I don't recall, but I think you need a 2/3 majority).  In my case, most teams felt that the cheating was obvious, but felt expulsion was too extreme.  So despite everyone encouraging me to take this up for vote, most didn't have the backbone to stand up to the cheater.  On a good note, he stopped sandbagging after that.

In the ABC rule book (old version), rule 17A covered unfair tactics, defined as intentionally bowling under a person's average to gain an advantage in league or tournament play.

I had to right a formal complaint to the President of the league, and list the charges.  I presented a very convincing argument with lots of data, and even a re-rate from a tournament.  There was literally no doubt he was cheating.  The most important factor is whether the other team captains will act, or not.

Good luck if you take that route, and be prepared for some backlash.  Doing what is right, isn't always popular.

This is a team effort for sandbagging not just one bowler.

The thought is when USBC is required to make a backbone decision they will not. Most in the league understand that, it is still sad that USBC will not do anything. (got to keep money coming in from your bowlers right?)

I like the idea of re-rating your averages. Say quarterly. Still has a team you can pick and choose who to slack off and when. No allowing them back seems the smarter choice.

I would think that since your a sanctioned league, contact the local USBC or State office and talk this over. Ask if one of them would visit the league during competition and then ask them for a direction. Also remind them that your paying for their support and expect something be done. In Dayton a few years ago we did this and  the bowler was without a(ABC card) for two years.
It can be done, but you need to continue to supply information, videos are easy these days and if possible last years records to see if that pattern is developing in the same manner. You will have to do the dirty work, as others will not, they will just complain and move on.
Also remind the rep from USBC, that next year the league could be unsanctioned as easy as being sanctioned. The state rep will be a lot more helpful.

There's a guy who was in our Jubilee League who was caught having 3 different USBC cards all with his first and last name spelled slightly different. Same person 3 different averages. He was the anchor on a team that's won the league the last 3 years professsional sandbaggers. A meeting was called and a USBC representative from our local association made this info known during the meeting and needles to say the team captains voted 27 to 1 to kick the guy out the league. Things have changed sandbagging is out of control. His teammate averaged 225 on Tuesday Night league and on the Wednesday Night Jubilee he averaged 181 same house same condition they kicked him out also.